On the computation of zone and double zone diagrams

Abstract

Classical objects in computational geometry are defined by explicit relations. Several years ago the pioneering works of T. Asano, J. Matousek and T. Tokuyama introduced "implicit computational geometry", in which the geometric objects are defined by implicit relations involving sets. An important member in this family is called "a zone diagram". The implicit nature of zone diagrams implies, as already observed in the original works, that their computation is a challenging task. In a continuous setting this task has been addressed (briefly) only by these authors in the Euclidean plane with point sites. We discuss the possibility to compute zone diagrams in a wide class of spaces and also shed new light on their computation in the original setting. The class of spaces, which is introduced here, includes, in particular, Euclidean spheres and finite dimensional strictly convex normed spaces. Sites of a general form are allowed and it is shown that a generalization of the iterative method suggested by Asano, Matousek and Tokuyama converges to a double zone diagram, another implicit geometric object whose existence is known in general. Occasionally a zone diagram can be obtained from this procedure. The actual (approximate) computation of the iterations is based on a simple algorithm which enables the approximate computation of Voronoi diagrams in a general setting. Our analysis also yields a few byproducts of independent interest, such as certain topological properties of Voronoi cells (e.g., that in the considered setting their boundaries cannot be "fat").

Very slight improvements (mainly correction of a few typos); add DOI; Ref [51] points to a freely available computer application which implements the algorithms; to appear in Discrete & Computational Geometry (available online)

References in corpus (7)

• The geometric stability of Voronoi diagrams with respect to small changes of the sites
• Distance k-Sectors Exist
• Zone diagrams in compact subsets of uniformly convex normed spaces
• Zone Diagrams in Euclidean Spaces and in Other Normed Spaces
• The geometric stability of Voronoi diagrams in normed spaces which are not uniformly convex
• Topological properties of sets represented by an inequality involving distances
• The trisector curve is transcendental

Cited by in corpus (5)

• Fixed points of polarity type operators
• The projector algorithm: a simple parallel algorithm for computing Voronoi diagrams and Delaunay graphs
• The geometric stability of Voronoi diagrams in normed spaces which are not uniformly convex
• Topological properties of sets represented by an inequality involving distances
• Stability of the optimal values under small perturbations of the constraint set